# geometric series

A geometric series is a series of the form

 $\displaystyle\sum_{i=1}^{n}ar^{i-1}$

(with $a$ and $r$ real or complex numbers). The partial sums of a geometric series are given by

 $\displaystyle s_{n}=\sum_{i=1}^{n}ar^{i-1}=\frac{a(1-r^{n})}{1-r}.$ (1)

An infinite geometric series is a geometric series, as above, with $n\rightarrow\infty$. It is denoted by

 $\displaystyle\sum_{i=1}^{\infty}ar^{i-1}$

If $|r|\geq 1$, the infinite geometric series diverges. Otherwise it converges to

 $\displaystyle\sum_{i=1}^{\infty}ar^{i-1}=\frac{a}{1-r}$ (2)

Taking the limit of $s_{n}$ as $n\rightarrow\infty$, we see that $s_{n}$ diverges if $|r|\geq 1$. However, if $|r|<1$, $s_{n}$ approaches (2).

One way to prove (1) is to take

 $\displaystyle s_{n}=a+ar+ar^{2}+\cdots+ar^{n-1}$

and multiply by $r$, to get

 $\displaystyle rs_{n}=ar+ar^{2}+ar^{3}+\cdots+ar^{n-1}+ar^{n}$

subtracting the two removes most of the terms:

 $\displaystyle s_{n}-rs_{n}=a-ar^{n}$

factoring and dividing gives us

 $\displaystyle s_{n}=\frac{a(1-r^{n})}{1-r}$

$\square$

Title geometric series GeometricSeries 2013-03-22 12:05:37 2013-03-22 12:05:37 mathcam (2727) mathcam (2727) 16 mathcam (2727) Definition msc 40A05 GeometricSequence ExampleOfAnalyticContinuation ApplicationOfCauchyCriterionForConvergence infinite geometric series